Best Known (33, 33+31, s)-Nets in Base 81
(33, 33+31, 486)-Net over F81 — Constructive and digital
Digital (33, 64, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- digital (16, 47, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- digital (2, 17, 116)-net over F81, using
(33, 33+31, 2363)-Net over F81 — Digital
Digital (33, 64, 2363)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8164, 2363, F81, 2, 31) (dual of [(2363, 2), 4662, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8164, 3286, F81, 2, 31) (dual of [(3286, 2), 6508, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8164, 6572, F81, 31) (dual of [6572, 6508, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 6573, F81, 31) (dual of [6573, 6509, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- linear OA(8161, 6562, F81, 31) (dual of [6562, 6501, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,15]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 6573, F81, 31) (dual of [6573, 6509, 32]-code), using
- OOA 2-folding [i] based on linear OA(8164, 6572, F81, 31) (dual of [6572, 6508, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(8164, 3286, F81, 2, 31) (dual of [(3286, 2), 6508, 32]-NRT-code), using
(33, 33+31, 8323641)-Net in Base 81 — Upper bound on s
There is no (33, 64, 8323642)-net in base 81, because
- 1 times m-reduction [i] would yield (33, 63, 8323642)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 1 716155 801847 639786 071800 192388 506321 910777 950126 546035 508967 660120 061593 679167 820896 784863 040479 354083 346298 462887 026401 > 8163 [i]